Responses to the Farmer MacDonald Problem
Hartweg, Kim, Teaching Children Mathematics
The problem appearing in the May 2003 "Problem Solvers" section was stated as follows:
Farmer MacDonald decided to make a graph to show the number of animals on his farm. He must do some mathematical calculations before he can prepare his graph. Here is what Farmer MacDonald knows about his animals:
* He has 12 cows.
* He has 1/2 as many pigs as cows.
* He has 4 more chickens than pigs.
* He has 4 more sheep than chickens.
Organize and display the data by creating a graph to represent Farmer MacDonald's animals. Explain how you organized the data.
The purpose of constructing a graph for the Farmer MacDonald problem is to give students the opportunity to become invested in the data and learn how graphs can display information in a variety of ways. Students in Erin Krieg's, Sue Campbell's, and Marlys Heisler's third-grade classes at Hamilton Elementary School in Hamilton, Illinois, were not given much guidance about how to construct the graph but were encouraged to create a graph that made sense to them so that they could explain the information to the class. After working with a partner to solve the problem and create a graph, students presented their work to the rest of the class and participated in a discussion about which graphs were easiest to read and understand.
Heisler's students represented the number of animals on Farmer MacDonald's farm in a variety of ways. Connor and Logan drew a grid to color squares (see fig. 1a), Jon and Scott drew squares and shaded them in (see fig. 1b), Erica and Shelby drew stars (see fig. 1c), Jacey and Kelly wrote out animal sounds (see fig. 1d), and Delaney and Jimmy numbered circles (see fig. 1e). After presenting their graphs to the class, the students decided that Connor and Logan's graph using the grid was the easiest for seeing the amount of each animal on Farmer MacDonald's farm because "they all started in the same place." Erica and Shelby's graph was the next graph of choice, but students decided that the number of stars in the first row should be consistent throughout the graph. When asked why Delaney and Jimmy drew their circles vertically for the pigs and horizontally for the cows, sheep, and chickens, Delaney stated, "We knew we would run out of room on our paper, so we decided to go across. …